#7198673

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996607
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
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1733 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

(2,4)(5,6,10,7)(8,11,15,13)(9,14,17,12)(16,19,20,18)(21,22,26,32,27,29,25,31)(23,33,24,30,34,36,35,28);(5,7,10,6)(8,11,15,13)(9,12,17,14)(16,19,20,18)(21,23)(22,28)(24,25)(26,35)(27,34)(29,30)(31,33)(32,36);(1,2,3,4)(5,8,10,15)(6,11,7,13)(9,16,17,20)(12,18,14,19)(22,29)(28,30)(31,32)(33,36);(8,15)(11,13)(16,20)(18,19)(21,24,25,23,27,35,26,34)(22,30,31,33,29,28,32,36);(5,9,10,17)(6,12,7,14)(8,16,15,20)(11,18,13,19)(21,25,27,26)(22,31,29,32)(23,24,34,35)(28,33,30,36)

Polycyclic presentation:

2^9|a^2=fh,b^2=h,c^2=ghi,d^2=fi,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308392, #72308638
maximal normal subgroup of	#72308392, #72308638

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7105869);