#129061

order	512
nilpotency class	2
isoclinism	isoclinic class of #7855
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₄²
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₂×#97
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

310 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=de,d^2=i,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,f^a=fi,c^b=ch,e^b=ei,f^c=fi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318459, #72318487, #72318536, …
maximal normal subgroup of	#72318459, #72318487, #72318536, …
maximal quotient of	#72322400

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36257);