#364566

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

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

2746 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315697, #72315713, #72315717
maximal normal subgroup of	#72315697, #72315713, #72315717

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271762);