#7627226

order	512
nilpotency class	2
isoclinism	isoclinic class of #63011
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×#3482	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7096 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:

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

Polycyclic presentation:

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

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, 7534422);