#357178

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#336 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1212 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)(5,6,8,7)(9,10,14,19,15,20,24,18)(11,21,23,12,13,17,22,16);(9,11,14,23,15,13,24,22)(10,16,19,21,20,12,18,17);(2,4)(5,6,8,7)(9,12,14,17,15,16,24,21)(10,13,19,22,20,11,18,23);(5,7,8,6)(9,13,15,11)(10,17,20,21)(12,19,16,18)(14,22,24,23)

Polycyclic presentation:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72240061, #72240067, #72240068

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264374);