#7065138

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2013 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=fg,c^2=h,d^2=fi,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308437, #72308732
maximal normal subgroup of	#72308437, #72308732

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6972334);