#7628689

order	512
nilpotency class	2
isoclinism	isoclinic class of #34980
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7910 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=h,d^2=h,f^2=i,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240254, #72240256, #72240258, …
maximal normal subgroup of	#72240254, #72240256, #72240258, …
maximal quotient of	#72280503, C₂×#7628689

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7535885);