#7626916

order	512
nilpotency class	2
isoclinism	isoclinic class of #62892
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³	C₂⁶
commutator	C₂²	C₂⁵×C₄
prime radical	C₂×#3521	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6628 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=g,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^d=eh

Conjugacy classes & character table

146 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534112);