#7622484

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622467
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂³×#99
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂×Q₈	C₂⁵
prime radical	#62852	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×Q₈ ◁ C₂²×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

4231 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340642
maximal normal subgroup of	#72340642
maximal quotient of	#72340719

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529680);