#129470

order	512
nilpotency class	4
isoclinism	isoclinic class of #129405
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#99
commutator	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1552 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 1552 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ch,d^b=di,f^b=fg,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268841
maximal normal subgroup of	#72268841
maximal quotient of	#72268853

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36666);