#265241

order	512
nilpotency class	3
isoclinism	isoclinic class of #265237
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#120	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1134 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1134 conjugacy classes of subgroups — too many to draw. Only its 158 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) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,d^2=eh,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,c^b=cg,f^b=fi,d^c=dh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339522, #72339526, #72339528
maximal normal subgroup of	#72339522, #72339526, #72339528

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 172437);