#7635072

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7885 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 20 points:

(5,6)(7,10)(8,9)(11,12)(13,14,17,15)(16,19,20,18);(2,4)(6,9)(10,12)(13,15,17,14)(16,19,20,18);(1,2)(3,4)(14,15)(18,19);(13,16,17,20)(14,18,15,19);(2,4);(5,7,8,11)(6,10,9,12)

Polycyclic presentation:

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

Relationships

maximal subgroup of	#72257534, #72257716, #72262511, …
maximal normal subgroup of	#72257534, #72257716, #72262511, …
maximal quotient of	C₂×#7635072, #72281808, #72282069

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7542268);