#7622485

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 ◁ C₂×#41 ◁ C₂×#585 ◁ 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:

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

Polycyclic presentation:

2^9|a^2=g,e^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	#72261053, #72261065, #72261176, …
maximal normal subgroup of	#72261053, #72261065, #72261176, …
maximal quotient of	#72318712

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529681);