#7203017

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200042
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#32911	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1593 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=ehi,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308700, #72308931, #72309019
maximal normal subgroup of	#72308700, #72308931, #72309019

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7110213);