#128288

order	512
nilpotency class	2
isoclinism	isoclinic class of #94871
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂³	C₂×#97
prime radical	C₂⁴	#97
LCS	G ▷ C₂²×C₄ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

350 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318580, #72318586, #72318592, …
maximal normal subgroup of	#72318580, #72318586, #72318592, …
maximal quotient of	#72322514

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35484);