#7056185

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
prime radical	C₂×#3267	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2034 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14,20,33,19,17,18,23)(15,24,34,21,26,39,27,31)(16,22,35,44,29,32,30,38)(25,40,43,36,28,41,42,37);(5,6)(7,11)(8,10)(9,12)(13,15,19,26)(14,21,17,31)(16,28,29,25)(18,34,20,27)(22,37,32,36)(23,39,33,24)(30,42,35,43)(38,40,44,41);(2,4)(5,7,8,12)(6,9,10,11)(13,16,19,29)(14,22,17,32)(15,25,26,28)(18,30,20,35)(21,36,31,37)(23,38,33,44)(24,40,39,41)(27,42,34,43);(7,12)(9,11)(13,17,20,23,19,14,18,33)(15,24,34,21,26,39,27,31)(16,22,35,44,29,32,30,38)(25,41,43,37,28,40,42,36);(6,10)(9,11)(14,17)(15,26)(16,29)(23,33)(27,34)(30,35)(36,37)(40,41)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=h,d^2=fi,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^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6963381);