#185068

order	512
nilpotency class	3
isoclinism	isoclinic class of #163899
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2392 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(5,6)(7,13,15,10)(8,11)(9,12)(14,19,20,17)(16,18)(22,26,28,31)(27,33,34,36);(5,7,9,15)(6,10,12,13)(8,14,16,20)(11,17,18,19)(21,22)(23,27)(24,31)(25,28)(26,32)(29,34)(30,36)(33,35);(2,4)(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(21,23)(22,27)(24,30)(25,29)(26,33)(28,34)(31,36)(32,35);(23,29)(27,34)(30,35)(33,36)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239068, #72239885
maximal normal subgroup of	#72239068, #72239885
maximal quotient of	#72239143, #72239900

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 92264);