C₂²×H(7,𝔽₂) #10587012

order	512
nilpotency class	2
isoclinism	isoclinic class of H(7,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁷

Aliases

direct product

C₂²×H(7,𝔽₂)

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂⁶
commutator, Frattini, cosocle	C₂	C₂⁸
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂ ▷ C₁

426010 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 8 generators on 20 points:

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

Conjugacy classes & character table

260 conjugacy classes. The character table is too large to display.

Relationships

Fitting subgroup of	#72233218, #72233726, #72258000, …
maximal subgroup of	#72233218, #72233689, D₄×H(7,𝔽₂), …
maximal normal subgroup of	#72233218, #72233689, D₄×H(7,𝔽₂), …
maximal quotient of	#72232884, C₂×#10586472, #72233491, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10494208);

C22×H(7,𝔽2) #10587012