#7628822

order	512
nilpotency class	2
isoclinism	isoclinic class of #35346
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	#35347	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8512 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)(7,10)(8,9)(11,12)(13,14,17,20)(15,18,22,26)(16,19,24,28)(21,25,23,27);(6,9)(10,12)(13,15)(14,18)(16,23)(17,22)(19,27)(20,26)(21,24)(25,28);(2,4)(13,16)(14,19)(15,21)(17,24)(18,25)(20,28)(22,23)(26,27);(16,24)(19,28)(21,23)(25,27);(15,22)(18,26)(21,23)(25,27);(5,7,8,11)(6,10,9,12)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72305286, #72315294
maximal normal subgroup of	#72305286, #72315294
maximal quotient of	#72306259

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7536018);