#7627130

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×#3552	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6802 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=gh,d^2=g,e^2=i,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, 7534326);