#7135208

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994043
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	C₂×D₄²	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1691 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:

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

Polycyclic presentation:

2^9|a^2=fgi,c^2=hi,d^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257290, #72257358, #72264310
maximal normal subgroup of	#72257290, #72257358, #72264310
maximal quotient of	#72308586

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7042404);