#366358

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2358
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1898 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)(5,6,10,14,11,15,9,24)(7,16,18,12,19,27,25,28)(8,13,22,30,23,32,21,31)(17,29,33,26,34,36,20,35);(5,7,11,19)(6,12,15,28)(8,20)(9,25,10,18)(13,29)(14,27,24,16)(17,22)(21,34)(23,33)(26,30)(31,35)(32,36);(2,4)(5,8)(6,13)(7,17)(9,21)(10,22)(11,23)(12,26)(14,30)(15,32)(16,29)(18,33)(19,34)(20,25)(24,31)(27,36)(28,35);(5,7)(6,12)(8,17)(9,25)(10,18)(11,19)(13,26)(14,27)(15,28)(16,24)(20,21)(22,33)(23,34)(29,31)(30,36)(32,35)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263012, #72280601
maximal normal subgroup of	#72263012, #72280601
maximal quotient of	#72263015

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 273554);