#246963

order	512
nilpotency class	3
isoclinism	isoclinic class of #246957
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#52008	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1237 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1237 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240219
maximal normal subgroup of	#72240219
maximal quotient of	#72240340, #72260896

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 154159);