#184762

order	512
nilpotency class	3
isoclinism	isoclinic class of #8304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂²×#525	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

4210 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,c^b=ci,g^b=gh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72266589, #72266590, #72314578, …
maximal normal subgroup of	#72266589, #72266590, #72314578, …
maximal quotient of	#72266599

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 91958);