#321776

order	512
nilpotency class	3
isoclinism	isoclinic class of #278923
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

692 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 692 conjugacy classes of subgroups — too many to draw. Only its 166 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) = 52
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 52 points:

(5,6)(7,15)(8,11)(9,19)(10,12)(13,17)(14,16)(18,20)(21,22,25,29,26,30,39,46)(23,27,33,42,32,41,49,52)(24,34,36,44,37,45,38,28)(31,47,48,43,35,51,50,40);(1,2,3,4)(5,7,16,18,10,17,9,8)(6,11,19,13,12,20,14,15)(21,23)(22,27)(24,35)(25,33)(26,32)(28,43)(29,42)(30,41)(31,37)(34,51)(36,50)(38,48)(39,49)(40,44)(45,47)(46,52);(2,4)(6,12)(11,20)(13,15)(14,19)(21,24,26,37)(22,28,30,44)(23,31,32,35)(25,38,39,36)(27,40,41,43)(29,45,46,34)(33,50,49,48)(42,51,52,47);(5,8,9,17,10,18,16,7)(6,13,14,11,12,15,19,20)(23,32)(27,41)(31,35)(33,49)(40,43)(42,52)(47,51)(48,50)

Polycyclic presentation:

2^9|a^2=fh,b^2=eghi,c^2=i,d^2=e,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=di,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 228972);