#7622911

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622911
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂²×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×Q₈	C₂⁵
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×D₄ ◁ C₂×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

3701 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)(3,11)(4,9)(5,8)(6,19)(7,12)(10,18)(13,25)(14,26)(15,23)(16,21)(17,29)(20,22)(24,28)(27,32)(30,31);(2,8)(3,6)(5,18)(9,21)(10,12)(11,22)(13,17)(16,28)(19,26)(23,24)(25,31)(29,32);(1,3)(4,13)(5,14)(6,7)(8,10)(11,19)(15,17)(16,27)(18,20)(21,24)(25,29)(28,30);(1,4)(2,9)(3,13)(5,16)(6,17)(7,15)(8,21)(10,24)(11,25)(12,23)(14,27)(18,28)(19,29)(20,30)(22,31)(26,32);(4,15)(9,23)(13,17)(16,28)(21,24)(25,29)(27,30)(31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261053, #72261107, #72261130
maximal normal subgroup of	#72261053, #72261107, #72261130

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530107);