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C₂³×#344 #6369766

order	512
nilpotency class	3
isoclinism	isoclinic class of #99
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂¹¹

Aliases

C₂³×#344

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂²⋊C₄
commutator	C₂²	C₂³×C₄²
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂⁴	#95
prime radical	C₂⁶	C₂×C₄
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

8266 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 22 points:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#123745, #72235161, C₂×#123492, …
maximal normal subgroup of	C₂×#123745, #72235161, C₂×#123492, …
maximal quotient of	#72235203, #72235218, #72235607, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6276962);