#7626340

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×#3545	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7226 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)(7,8)(9,10)(11,12)(14,19)(17,25)(20,26)(24,28);(6,10)(7,11)(16,23)(20,26)(22,27)(24,28);(5,7,9,11)(6,8,10,12)(13,14,18,19)(15,17,21,25)(16,20,23,26)(22,24,27,28);(1,2,4,3)(5,8)(6,11)(7,10)(9,12)(13,15,18,21)(14,17,19,25)(16,22,23,27)(20,24,26,28);(1,3,4,2)(13,16)(14,20)(15,22)(17,24)(18,23)(19,26)(21,27)(25,28);(13,17)(14,21)(15,19)(16,24)(18,25)(20,27)(22,26)(23,28)

Polycyclic presentation:

2^9|a^2=i,c^2=gh,d^2=hi,e^2=i,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, 7533536);