#7021641

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2324 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257034, #72257130, #72257134
maximal normal subgroup of	#72257034, #72257130, #72257134
maximal quotient of	#72307785

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6928837);