#323452

order	512
nilpotency class	3
isoclinism	isoclinic class of #298661
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

724 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 226 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=efh,b^2=egi,c^2=i,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^c=fh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224775, #72224779, #72224781
maximal normal subgroup of	#72224775, #72224779, #72224781

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 230648);