#7635000

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
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
prime radical	#37673	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7292 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dh,c^b=ci,e^c=ehi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72305286
maximal normal subgroup of	#72305286
maximal quotient of	#72306261

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7542196);