#246988

order	512
nilpotency class	3
isoclinism	isoclinic class of #246987
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1431 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1431 conjugacy classes of subgroups — too many to draw. Only its 150 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(2,5)(3,7)(6,8)(9,10)(11,21)(12,26)(13,19)(14,18)(15,20)(16,23)(17,27)(22,31)(24,33)(25,32)(28,35)(29,34)(30,38)(36,37)(39,40);(3,7)(5,8)(9,11,15,25)(10,16,20,33)(12,22,28,38)(13,24,14,23)(17,31,34,30)(18,32,19,21)(26,36,35,40)(27,39,29,37);(1,2)(3,8)(4,6)(5,7)(9,12)(10,17)(11,22)(13,27)(14,29)(15,28)(16,30)(18,26)(19,35)(20,34)(21,36)(23,37)(24,39)(25,38)(31,33)(32,40);(3,7)(5,8)(9,12)(10,17)(11,22)(13,29)(14,27)(15,28)(16,31)(18,35)(19,26)(20,34)(21,36)(23,39)(24,37)(25,38)(30,33)(32,40)

Polycyclic presentation:

2^9|b^2=h,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,g^a=gh,c^b=cg,e^b=eh,f^b=fhi,e^c=ei,f^c=fhi,e^d=eh,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340236, #72340417, #72340421
maximal normal subgroup of	#72340236, #72340417, #72340421

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 154184);