#7626301

order	512
nilpotency class	2
isoclinism	isoclinic class of #62817
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×#3494	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7274 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)(6,8)(10,12)(13,14)(15,16)(17,25)(18,19)(20,23)(21,26)(22,27)(24,28);(7,11)(10,12)(14,19)(16,20)(21,25)(24,27);(1,3,4,2)(5,6,9,8)(7,10,11,12)(13,14,18,19)(15,20,23,16)(17,21,26,25)(22,27,28,24);(13,15)(14,20)(16,19)(17,22)(18,23)(21,27)(24,25)(26,28);(5,7)(6,10)(8,12)(9,11)(13,16,18,20)(14,15,19,23)(17,24,26,27)(21,22,25,28);(5,8,9,6)(7,12,11,10)(13,17,18,26)(14,21,19,25)(15,22,23,28)(16,24,20,27)

Polycyclic presentation:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7533497);