#378037

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1251 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=ei,d^2=fh,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gi,d^c=di,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226917, #72226919, #72226937
maximal normal subgroup of	#72226917, #72226919, #72226937

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 285233);