#356504

order	512
nilpotency class	3
isoclinism	isoclinic class of #22455
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1948 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=g,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,d^b=de,g^b=gi,f^c=fi,e^d=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251058, #72251086, #72251089
maximal normal subgroup of	#72251058, #72251086, #72251089
maximal quotient of	#72314517

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263700);