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C₂×#7555 #153732

order	512
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂¹²

Aliases

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂³
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂⁷	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

19546 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 22 points:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72238083, #72267766, #72267784
Frattini subgroup of	#72238081, #72238083, #72238091, …
Fitting subgroup of	#72255760, #72261990
maximal subgroup of	C₂×#106076, #72219227, #72219230, …
maximal normal subgroup of	C₂×#106076, #72219227, #72219230, …
maximal quotient of	C₂×#123640, #72235046, #72238285, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60928);