#7622497

order	512
nilpotency class	4
isoclinism	isoclinic class of #62636
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

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	C₂²×#576	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4084 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,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	#72226792, #72226794, #72226798, …
maximal normal subgroup of	#72226792, #72226794, #72226798, …
maximal quotient of	D₄×#2140

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529693);