#7635040

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	#37649	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7137 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)(13,14,17,20)(15,21,23,18)(16,24,25,19)(22,26,28,27);(2,4)(6,9)(10,12);(1,2)(3,4)(13,15)(14,18)(16,22)(17,23)(19,26)(20,21)(24,27)(25,28);(13,16,17,25)(14,19,20,24)(15,22,23,28)(18,26,21,27);(2,4)(15,23)(18,21)(22,28)(26,27);(5,7,8,11)(6,10,9,12)

Polycyclic presentation:

2^9|a^2=h,d^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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7542236);