#7626693

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×#3505	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7254 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)(6,9)(10,12)(14,19)(15,20)(17,25)(24,28);(1,3,4,2)(13,14)(15,22)(16,20)(17,21)(18,19)(23,28)(24,27)(25,26);(5,6)(7,10)(8,9)(11,12)(13,15,18,20)(14,16,19,22)(17,23,25,27)(21,24,26,28);(7,11)(10,12)(13,14)(15,16)(17,21)(18,19)(20,22)(23,24)(25,26)(27,28);(5,7,8,11)(6,10,9,12)(13,16)(14,20)(15,19)(17,24)(18,22)(21,27)(23,26)(25,28);(5,6)(7,10)(8,9)(11,12)(13,17)(14,21)(15,23)(16,24)(18,25)(19,26)(20,27)(22,28)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,e^2=gh,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, 7533889);