#7622614

order	512
nilpotency class	4
isoclinism	isoclinic class of #62636
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator	C₈	C₂⁴×C₄
Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	C₂×#3414	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1954 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=i,f^2=hi,h^2=i,b^a=bf,e^a=ei,f^a=fh,h^a=hi,f^b=fh,h^b=hi,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320360, #72320393, #72320537
maximal normal subgroup of	#72320360, #72320393, #72320537

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529810);