#146471

order	512
nilpotency class	4
isoclinism	isoclinic class of #11508
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#99
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂³×Q₈	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁵ ◁ C₂⁴×Q₈ ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1690 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1690 conjugacy classes of subgroups — too many to draw. Only its 210 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) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,h^a=hi,c^b=ci,f^b=fhi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72266764, #72266766, #72266768, …
maximal normal subgroup of	#72266764, #72266766, #72266768, …
maximal quotient of	#72266775, #72267713

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53667);