#7627193

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×#3541	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7018 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,d^2=h,e^2=hi,f^2=h,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, 7534389);