#7634882

order	512
nilpotency class	2
isoclinism	isoclinic class of #36702
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

8882 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)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,19)(16,24)(17,22)(18,21)(20,25)(23,28)(26,27);(2,4)(5,7,8,11)(6,9,10,12)(13,15,18,17)(14,19,21,22)(16,23,25,26)(20,27,24,28);(6,10)(7,11)(13,16)(14,20)(15,23)(17,26)(18,25)(19,27)(21,24)(22,28);(6,10)(9,12)(14,21)(16,25)(19,22)(23,26);(14,21)(19,22)(20,24)(27,28);(13,17,18,15)(14,22,21,19)(16,26,25,23)(20,28,24,27)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282531, #72312926, #72314942
maximal normal subgroup of	#72282531, #72312926, #72314942
maximal quotient of	#72283268

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7542078);