#354817

order	512
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3706 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,e^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,e^b=ei

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320920, #72321037
maximal normal subgroup of	#72320920, #72321037
maximal quotient of	#72321229

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262013);